1. Warm Up Identify the domain and range of each function.
1. f(x) = x2 + 2
D: R; R:{y|y ≥2}
2. f(x) = 3x3
D: R; R: R
Use the description to write the quadratic function g based on the parent function f(x) = x2.
3. f is translated 3 units up.
g(x) = x2 + 3
4. f is translated 2 units left.
g(x) =(x + 2)2

2. Objectives Graph radical functions
Transform radical functions by changing parameters.

3. Check It Out! Example 1
Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x if possible.

4. x
(x, f(x)) –1 1 4 9

6. Check It Out! Example 2
Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x if possible.

7. Check It Out! Example 2 Continued
(x, f(x)) –8
(–8, –2) –1
(–1,–1)
(0, 0) 1
(1, 1) 8
(8, 2) • • • • •
The domain is the set of all real numbers. The range is also the set of all real numbers.

8. Check It Out! Example 3
Graph each function, and identify its domain and range. x
(x, f(x)) –1 3 8 15

9. • • • • x (x, f(x)) Check It Out! Example 3
Graph each function, and identify its domain and range. x
(x, f(x)) –1
(–1, 0) 3
(3, 2) 8
(8, 3) 15
(15, 4) • • • •
The domain is {x|x ≥ –1}, and the range is {y|y ≥0}.

10. The graphs of radical functions can be transformed by using methods similar to those used to transform linear, quadratic, polynomial, and exponential functions. This lesson will focus on transformations of square-root functions.

12. Check It Out! Example 4
Using the graph of as a guide, describe the transformation and graph the function.
f(x)= x •
g(x) = x + 1
Translate f 1 unit up.

13. g is f vertically compressed by a factor of .
Check It Out! Example 5
Using the graph of as a guide, describe the transformation and graph the function.
f(x) = x
g is f vertically compressed by a factor of 1 2

14. General Equation The general form of the square root function is
The cube root function is

15. Try these!
Using the graph of as a guide, describe the transformation and graph the function.
f(x)= x ●
g(x) = –3 x – 1
g is f vertically stretched by a factor of 3, reflected across the x-axis, and translated 1 unit down.

16. Check It Out! Example 6
Use the description to write the square-root function g.
The parent function is reflected across the x-axis, stretched vertically by a factor of 2, and translated 1 unit up.
f(x)= x

17. Check It Out! Example 6 Solution
Use the description to write the square-root function g.
The parent function is reflected across the x-axis, stretched vertically by a factor of 2, and translated 1 unit up.
f(x)= x
Step 1 Identify how each transformation affects the function.
Reflection across the x-axis: a is negative
a = –2
Vertical compression by a factor of 2
Translation 1 unit up: k = 1

18. Check It Out! Example 6 Continued
Step 2 Write the transformed function.
Substitute –2 for a and 1 for k.
Simplify.
Check Graph both functions on a graphing calculator. The g indicates the given transformations of f.